06 Exponential and logarithmic functions · g10-06-01 · 5–10 min
A negative exponent is a reciprocal
Does 2⁻³ mean a negative number or a reciprocal?
Predict: Continue the sequence 2³,2²,2¹,2⁰ one step at a time to exponent −3.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base a = 2; Integer exponent n = -3 | |
Case 2
Base a = -2; Integer exponent n = -3 | |
Case 3
Base a = 0; Integer exponent n = -1 | |
Revise: My claim needs the condition… because…
Exit: Find 5⁻².
06 Exponential and logarithmic functions · g10-06-02 · 5–10 min
A fractional exponent names a root
What does 16^(3/4) ask you to do?
Predict: Take a fourth root and cube it; predict whether reversing those two operations changes the result.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Positive base a = 16; Integer numerator p = 3; Positive denominator q = 4 | |
Case 2
Positive base a = 16; Integer numerator p = -3; Positive denominator q = 4 | |
Case 3
Positive base a = 1; Integer numerator p = 3; Positive denominator q = 4 | |
Revise: My claim needs the condition… because…
Exit: Evaluate 27^(2/3).
06 Exponential and logarithmic functions · g10-06-03 · 5–10 min
Equal additions or equal ratios?
Does adding 20 each round match growing by 20% each round?
Predict: Starting at 100, predict both quantities after two rounds before reading the graph.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Multiplier r = 1.2; Number of rounds n = 2 | |
Case 2
Multiplier r = 0.8; Number of rounds n = 3 | |
Case 3
Multiplier r = 1; Number of rounds n = 3 | |
Revise: My claim needs the condition… because…
Exit: After three rounds retaining 80% of 100, what remains?
06 Exponential and logarithmic functions · g10-06-04 · 5–10 min
Crossing the base 1 reverses growth
What changes in aˣ when a passes through 1?
Predict: Compare outputs at x=−1,0,1 for bases 0.5,1,2.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base a = 2; Input x = 1 | |
Case 2
Base a = 0.5; Input x = 1 | |
Case 3
Base a = 1; Input x = 1 | |
Revise: My claim needs the condition… because…
Exit: Is (1/2)ˣ increasing or decreasing, and can its value be zero?
06 Exponential and logarithmic functions · g10-06-05 · 5–10 min
A logarithm asks for an exponent
What number does log₂8 ask for, and why is log₂0 impossible?
Predict: Complete 2^?=8 and 2^?=1/4 before selecting logarithm inputs.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Logarithm base a = 2; Argument b = 8 | |
Case 2
Logarithm base a = 2; Argument b = 0.25 | |
Case 3
Logarithm base a = 2; Argument b = 0 | |
Revise: My claim needs the condition… because…
Exit: Evaluate log₃(1/9).
06 Exponential and logarithmic functions · g10-06-06 · 5–10 min
Swap coordinates for inverse graphs
If (2,4) lies on y=2ˣ, which point lies on y=log₂x?
Predict: Reflect the point across y=x before plotting its inverse partner.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base a = 2; Exponential input x = 2 | |
Case 2
Base a = 2; Exponential input x = -2 | |
Case 4
Base a = 1; Exponential input x = 1 | |
Revise: My claim needs the condition… because…
Exit: Reflect (−2,1/4) from y=2ˣ into its inverse graph.
06 Exponential and logarithmic functions · g10-06-07 · 5–10 min
Products become sums of logarithms
Is logₐ(u+v)=logₐu+logₐv?
Predict: For base 2 and u=v=2, predict the logarithms of the product and of the sum.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base a = 2; First argument u = 2; Second argument v = 2 | |
Case 2
Base a = 2; First argument u = 2; Second argument v = 4 | |
Case 3
Base a = 2; First argument u = 8; Second argument v = 2 | |
Revise: My claim needs the condition… because…
Exit: Evaluate log₂8−log₂2 and explain the single-log form.
06 Exponential and logarithmic functions · g10-06-08 · 5–10 min
The threshold between two integer rounds
When does 100rⁿ first reach a target, and why might rounding a logarithm fail?
Predict: For growth r=1.2 and target 150, predict the first whole round that reaches the target.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Multiplier r = 1.2; Positive target = 150 | |
Case 2
Multiplier r = 0.8; Positive target = 50 | |
Case 3
Multiplier r = 1.2; Positive target = 100 | |
Revise: My claim needs the condition… because…
Exit: 100×1.2ⁿ reaches at least 150 first at which integer n?
06 Exponential and logarithmic functions · g10-06-09 · 5–10 min
A decreasing base reverses the inequality
Does logₐx<logₐb always imply x<b?
Predict: Solve log₂x<log₂4 and log_(1/2)x<log_(1/2)4 before testing a value.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base a = 2; Test input x = 2; Comparison input b = 4 | |
Case 2
Base a = 0.5; Test input x = 2; Comparison input b = 4 | |
Case 3
Base a = 0.5; Test input x = 4; Comparison input b = 4 | |
Revise: My claim needs the condition… because…
Exit: Solve log_(1/2)x<log_(1/2)4.
